Intuitionistic Logic (Wikipedia
[accessed 19-Jul-2015], Stanford
Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be
thought of as a constructive logic in which we must build and exhibit
concrete examples of objects before we can accept their existence.
Unproved statements in intuitionistic logic are not given an intermediate truth value,
instead, they remain of unknown truth value until they are either proved or disproved.
Intuitionist logic can also be thought of as a weakening of classical
logic such that the law of excluded middle (LEM), (φ ∨
¬ φ), doesn't always hold.
Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if
we don't have a proof of either one.
There is also no rule for double negation elimination.
Brouwer observed in 1908 that LEM was abstracted from finite situations,
then extended without justification to statements about infinite collections.

Mario Carneiro's work (Metamath database)
"iset.mm" provides in Metamath a development of
"set.mm" whose eventual
aim is to show how many of the theorems of set theory and
mathematics that can be derived from classical first order logic can
also be derived from a weaker system called "intuitionistic logic." To
achieve this task, iset.mm adds (or substitutes) 12 intuitionistic
axioms whose second part of the name begins with the letter "i" to the
classical logical axioms of set.mm.

Among these 12 new axioms, the 6 first
(ax-ia1,
ax-ia2,
ax-ia3,
ax-io,
ax-in1
and
ax-in2), when substituted to
ax-3 and added with
ax-1,
ax-2
and
ax-mp,
allow for the development of intuitionistic
propositional logic. Each of them is a theorem of classical
propositional logic, but ax-3 cannot be derived from them. Similarly,
other basic classical theorems, like the third middle excluded or the
equvalence of a proposition with its double negation, cannot be derived
in intuitionistic propositional calculus. Glivenko showed that a
proposition φ
is a theorem of classical propositional calculus if and only
if ¬¬φ
is a theorem of intuitionistic propositional calculus.

The last two new axioms
(ax-i9
and
ax-i12)
are variants of the
classical axioms
ax-9
and
ax-12.
The substitution of ax-i9 and
ax-i12 with ax-9 and ax-12 and the inclusion of
ax-8,
ax-10,
ax-11,
ax-13,
ax-14
and
ax-17
allow for the development of the
intuitionistic predicate calculus.

Each of the 12 new axioms is a theorem of classical first order
logic with equality. But some axioms of classical first order logic
with equality, like ax-3, cannot be derived in the intuitionistic
predicate calculus.

One of the major interests of the intuitionistic predicate calculus
is that its use can be considered as a realization of the program of the
constructivist philosophical view of mathematics.

The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if
φ implies a contradiction (such as its own negation),
then one can conclude ¬ φ. By contrast, assuming
¬ φ
and then deriving a contradiction only serves to prove ¬ ¬ φ,
which in intuitionistic logic is not the same as φ.

The biconditional can be defined as the conjunction of two implications, as in
dfbi2 and df-bi. Other ways of understanding
the biconditional, such as dfbi1 or dfbi3,
however, are classical rather than intuitionistic results (following the proofs on those pages shows they depend on ax-3).

Predicate
logic adds set variables (individual variables) and the ability to quantify
them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃
cannot be defined in terms of ∀. As in classical logic, we also add = for equality
(which is key to how we handle substitution in metamath) and ∈ (which for current
purposes can just be thought of as an arbitrary predicate, but which will later come to
mean set membership).

Our axioms are based on the classical set.mm predicate logic axioms, but adapted for
intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing
some aspects of how we handle negations.

[Lopez-Astorga] Lopez-Astorga, Miguel,
"The First Rule of Stoic Logic and its Relationship with the
Indemonstrables", Revista de Filosofía Tópicos (2016); available
at
http://www.scielo.org.mx/pdf/trf/n50/n50a1.pdf (retrieved 3 Jul
2016).

[Megill] Megill, N.,
"A Finitely Axiomatized Formalization of Predicate Calculus with
Equality," Notre Dame Journal of Formal Logic, 36:435-453
(1995) [QA.N914]; available at http://projecteuclid.org/euclid.ndjfl/1040149359 (accessed
11 Nov 2014); the PDF
preprint has the same content (with corrections) but pages are
numbered 1-22, and the database references use the numbers printed on the
page itself, not the PDF page numbers.